Abstract
We present a methodology for finding globally optimal knockout tournament designs when partial information is known about the strengths of the players. Our approach involves maximizing an expected utility through a Bayesian optimal design framework. Given the prohibitive computational barriers connected with direct computation, we compute a Monte Carlo estimate of the expected utility for a fixed tournament bracket, and optimize the expected utility through simulated annealing. We demonstrate our method by optimizing the probability that the best player wins the tournament. We compare our approach to other knockout tournament designs, including brackets following the standard seeding. We also demonstrate how our approach can be applied to a variety of other utility functions, including whether the best two players meet in the final, the consistency between the number of wins and the player strengths, and whether the players are matched up according to the standard seeding.
References
Bradley, R. A. and M. E. Terry. 1952. “Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons.” Biometrika 39(3/4):324–345.10.1093/biomet/39.3-4.324Search in Google Scholar
David, H. 1988. The Method of Paired Comparisons. London: Charles Griffin.Search in Google Scholar
Genz, A. 1992. “Numerical Computation of Multivariate Normal Probabilities.” Journal of Computational and Graphical Statistics 1:141–149.Search in Google Scholar
Glickman, M. E. 2008. “Bayesian Locally Optimal Design of Knockout Tournaments.” Journal of Statistical Planning and Inference 138:2117–2127.10.1016/j.jspi.2007.09.007Search in Google Scholar
Glickman, M. E. and S. T. Jensen. 2005. “Adaptive Paired Comparison Design.” Journal of Statistical Planning and Inference 127:279–293.10.1016/j.jspi.2003.09.022Search in Google Scholar
Hwang, F. 1982. “New Concepts in Seeding Knockout Tournaments.” American Mathematical Monthly 89(4):235–239.10.1080/00029890.1982.11995420Search in Google Scholar
Johnson, D. S., C. R. Aragon, L. A. McGeoch, and C. Schevon. 1989. “Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning.” Operations Research 37:865–892.10.1287/opre.37.6.865Search in Google Scholar
Johnson, D. S., C. R. Aragon, L. A. McGeoch, and C. Schevon. 1991. “Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning.” Operations Research 39:378–406.10.1287/opre.39.3.378Search in Google Scholar
Kirkpatrick, S., C. D. Gelatt, and M. P. Vecchi. 1983. “Optimization by simulated Annealing.” Science 220:671–680.10.1126/science.220.4598.671Search in Google Scholar PubMed
Lindley, D. 1972. Bayesian Statistics: A Review. Philadelphia: SIAM.10.1137/1.9781611970654Search in Google Scholar
Morokoff, W. J. and R. E. Caflisch. 1995. “Quasi-Monte Carlo Integration.” Journal of Computational Physics 122:218–230.10.1006/jcph.1995.1209Search in Google Scholar
Mosteller, F. 1951. “Remarks on The Method of Paired Comparisons: I. The Least Squares Solution Assuming Equal Standard Deviations and Equal Correlations.” Psychometrika 16:3–9.Search in Google Scholar
Sacks, J. and S. Schiller. 1988. “Spatial Designs.” Statistical Decision Theory and Related Topics IV 2:385–399.10.1007/978-1-4612-3818-8_32Search in Google Scholar
Schwenk, A. J. 2000. “What is The Correct Way to Seed a Knockout Tournament?” American Mathematical Monthly 107(2):140–150.10.1080/00029890.2000.12005171Search in Google Scholar
Searls, D. T. 1963. “On the Probability of Winning with Different Tournament Procedures.” Journal of the American Statistical Association 58:1064–1081.10.1080/01621459.1963.10480688Search in Google Scholar
Thurstone, L. L. 1927. “A Law of Comparative Judgment.” Psychological Review 34:273.10.1037/h0070288Search in Google Scholar
©2016 by De Gruyter